Details

Giovanni Cimatti ^[a]

Existence and non-existence of functional solutions for the equations of axially symmetric gravitational fields in general relativity

Pages: 287-306
Received: 14 September 2016
Accepted in revised form: 20 June 2017
Mathematics Subject Classification (2010): 83C10, 83C05.
Keywords: Axially symmetric gravitational fields, existence and uniqueness of solutions, Weyl-Lewis-Papapetrou coordinates.
Author address:
[a]: Department of Mathematics, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Abstract: A theorem of existence and non-existence of solutions for a boundary value problem for the equations of axially symmetric gravitational field in vacuum is given using the method of functional solutions. The boundary value problem is reduced to a two-point problem for a Bernoulli equation. Conditions are given under which solutions exist or not exist.

References

[1]: J. Bičák, Selected solutions of Einstein's field equations: their role in general relativity and astrophysics, in "Einstein's field equations and their physical implications", Bern G. Schmidt, ed., Lecture Notes in Phys., 540, Springer-Verlag, Berlin, 2000, 1-126. MR1765129
[2]: A. C. Gutiérrez-Piñeres, G. A. González and H. Quevedo, Conformastatic disk-haloes in Einstein-Maxwell gravity, Phys. Rev. D 87 (2013), 044010. DOI: 10.1103/PhysRevD.87.044010
[3]: A. Einstein, Cosmological considerations on the general theory of relativity, in "The Principle of Relativity", Dover, London, 1952.
[4]: L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Pergamon Press, Paris-Frankfurt, 1962. MR0143451 zbMATH
[5]: T. Lewis, Some special solutions of the equations of axially symmetric gravitational fields, Proc. R. Soc. Lond., Ser. A 136 (1932), 176-192. zbMATH
[6]: R. A. Matzner and C. W. Misner, Gravitational field equations for sources with axial symmetry and angular momentum, Phys. Rev. 154 (1967), 1229-1232. DOI: 10.1103/PhysRev.154.1229
[7]: A. Papapetrou, Eine rotationssymmetrische Lösung in der allgemeinen Relativitätstheorie, Ann. Physik 12 (1953), 309-315. MR0058339
[8]: A. Papapetrou, Champs gravitationnelles stationnaries à symétrie axiale, Annales de l'I.H.P. Physique théorique 4 (1966), 83-105. Stable URL
[9]: H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein's field equations, Cambridge Univ. Press, Cambridge, 2003. MR2003646
[10]: H. Weyl, Zur Gravitationstheorie, Ann. der Physik 54 (1917), 117-145. zbMATH